What is the integral of (10)/((25x^2+1)^2) ?

The integral of (10)/((25x^2+1)^2) is arctan(5x)+1/2 sin(2arctan(5x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (10)/((25x^2+1)^2)

\int \frac{10}{( 25 x ^{2} + 1 ) ^{2}} d x

arctan (5 x) + \frac{1}{2} sin (2 arctan (5 x)) + C

Solution steps

\int \frac{10}{( 25 x ^{2} + 1 ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int \frac{1}{( 25 x ^{2} + 1 ) ^{2}} d x

Apply Integral Substitution

= 10 \cdot \int \frac{1}{5 ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \frac{1}{5} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply Trigonometric Substitution

: \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

= 10 \cdot \frac{1}{5} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Rewrite using trig identities

= 10 \cdot \frac{1}{5} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \frac{1}{5} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 10 \cdot \frac{1}{5} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= 10 \cdot \frac{1}{5} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= 10 \cdot \frac{1}{5} \cdot \frac{1}{2} (arctan (5 x) + \frac{1}{2} sin (2 arctan (5 x)))

Simplify

10 \cdot \frac{1}{5} \cdot \frac{1}{2} (arctan (5 x) + \frac{1}{2} sin (2 arctan (5 x))) : arctan (5 x) + \frac{1}{2} sin (2 arctan (5 x))

= arctan (5 x) + \frac{1}{2} sin (2 arctan (5 x))

Add a constant to the solution

= arctan (5 x) + \frac{1}{2} sin (2 arctan (5 x)) + C

\int_{1}^{5} 7 r^{2} ln (r) d r

x \to - 2 lim (\frac{3}{x ^{2} - 4})

\int \frac{x - 2}{x ^{3}} d x

\frac{d}{d x} ((\frac{- 2 x - 1}{- 4 x + 2})^{5})

\int_{2}^{4} 2 x^{2} - 3 x + 1 d x

What is the integral of (10)/((25x^2+1)^2) ?
The integral of (10)/((25x^2+1)^2) is arctan(5x)+1/2 sin(2arctan(5x))+C